CN103235888B - The method of a kind of accurate Calculation Bianisotropic medium ball electromagnetic scattering - Google Patents

Abstract

The present invention proposes the method for a kind of accurate Calculation Bianisotropic medium ball electromagnetic scattering.Step of the present invention is as follows: 1. the differential equation utilizing the eigen[value derivation magnetic induction density <b>B</bGreatT.Gr eaT.GT of passive maxwell equation group and two anisotropy medium; 2. the factor relevant with <b>B</bGreatT.Gr eaT.GT in the differential equation is expressed with the form of spherical vector wave functions, then spherical vector wave functions <b>M is utilized, the property of orthogonality of N</b> draws a matrix equation containing ginseng, the condition first utilizing matrix equation to meet untrivialo solution calculates the parameter of this matrix equation, then parameter generation is got back to the untrivialo solution obtaining matrix equation in the matrix equation of containing parameter; Does 3. the function that structure one is new, use new function again magnetic induction density <b>B is represented, </b> and then obtain the electromagnetic field of medium ball inside, then the electromagnetic field in medium ball and the incident electromagnetic field outside ball, scattering field are updated in boundary condition, draw scattering matrix.The present invention is applicable to the electromagnetic scattering solving the less Bianisotropic medium ball of electric size.

Description

The method of a kind of accurate Calculation Bianisotropic medium ball electromagnetic scattering

Technical field

The invention belongs to Electromagnetic Scattering Theory and calculate field, be specifically related to the method for a kind of accurate Calculation Bianisotropic medium ball electromagnetic scattering.

Background technology

Solving the more traditional research method of electromagnetic scattering is exactly analytical method, and this is the target that scientist pursues always.So-called analysis research method is a kind of method of mathematical solution of closing form, is directly ask the various math equations of being derived by maxwell equation group.Electromagnetic problem for some boundary condition rule is very effective.Analytic solution can provide more effective data for other numerical evaluation, and the correctness of logarithm value result of calculation is verified, and can provide physical concept clearly, thus have very important guiding significance.

The mediums such as the plasma under the effect of stationary magnetic field and ferrite, their electromagnetic property will use tensor dielectric coefficient respectively and tensor permeability describe, namely , they have anisotropic character, are called anisotropy ature of coal.

Research for anisotropic medium ball electromagnetic scattering analytic solution is carried out more extensive now, also can calculate the RCS of ball with different analytic methods.

Two anisotropy medium makes the cross-couplings between the electric field that provides and magnetic field, for the analytic solution solving this kind of medium ball become very difficult.So the research of Bianisotropic medium ball electromagnetic scattering analytic solution does not seldom almost have.

Summary of the invention

The object of the invention is to for the deficiencies in the prior art, propose the method for a kind of accurate Calculation Bianisotropic medium ball electromagnetic scattering, prove that spherical vector wave functions is applicable to Bianisotropic medium simultaneously.

The technical solution adopted for the present invention to solve the technical problems is as follows:

Step 1. utilizes the eigen[value of passive maxwell equation group and two anisotropy medium to derive about magnetic induction density bthe differential equation;

The differential equation neutralizes by step 2. bthe relevant factor is expressed with the form of spherical vector wave functions, then utilizes spherical vector wave functions m,Nproperty of orthogonality draw the matrix equation of a containing parameter, the condition first utilizing matrix equation to meet untrivialo solution calculates the parameter of this matrix equation, then parameter generation is got back to the untrivialo solution obtaining matrix equation in the matrix equation of containing parameter;

Step 3. constructs a new function, uses new function again magnetic induction density is represented b,and then obtain the electromagnetic field of medium ball inside, then the electromagnetic field in medium ball and the incident electromagnetic field outside ball, scattering field are updated in boundary condition, draw scattering matrix.

As described in step 1, become two anisotropy medium by adding one in anisotropy medium eigen[value, the eigen[value of two anisotropy medium is specific as follows:

(1)

Wherein, electric displacement vector d, electric field intensity e, magnetic field intensity hand magnetic induction density bbe all vector, represent vector with black runic; represent imaginary unit; be used to the parameter weighing medium electromagnetic property;

Passive maxwell equation group is specific as follows:

(2a)

(2b)

(2c)

(2d)

Wushu 1 is updated in formula 2a, 2b, 2c, 2d, derives magnetic induction density bthe differential equation as follows:

(3)

Wherein, symbol ▽ × represent and curl is asked to a vector; ω is electromagnetic frequency; for inverse; ;

As described in step 2, by formula 3 write as the form of spherical vector wave functions, specific as follows:

(4)

(5)

(6)

Wherein, with represent and equally represent spherical vector wave functions, subscript (1) represents that vector wave function is made up of first kind spheric Bessel function, subscript represent the parameter in spherical vector wave functions; represent that one is treated quantitatively, represent a vector in spherical coordinate system; Coefficient before spherical vector wave functions is determined by the tensor in medium eigen[value, and , represent the field intensity of incident electric fields.

Coefficient definition before vector wave function function is specific as follows:

， (7)

， (8)

(9)

(10)

(11)

(12)

(13)

Wherein: , when time, ; When time, , ;

， (14)

， (15)

(16)

， (17)

(18)

Wherein, in the present invention, represent same amount, lower target difference is in order to distinguish in different expression formulas; In like manner , , all represent same amount.

Formula 4,5,6 is updated in formula 3 and obtains:

(19)

The character of spherical vector wave functions is utilized to obtain:

Be expressed as follows by the form of matrix:

(20)

Formula 20 changes following form into:

(21)

Wherein , Ibe unit matrix, the implication that formula 21 is expressed is: there is such parameter k and make equation have untrivialo solution, know that the determinant that only need make formula 21 is zero, solve parameter by matrix knowledge , parameter note solution is , then use substitution formula 20 obtains the non-vanishing solution of equation , note ,

A new phasor function is constructed in described step 3 , specific as follows:

Wherein for undetermined coefficient, determined by the boundary condition of medium ball surface;

Order (22)

(23)

(24)

The subscript that coefficient in formula 22,23,24 before vector wave function has more lto separate cause in substitution formula 7,8,9,16; , , all non-vanishing number, and the only vanishing when asking curl to them.

parameter is wherein specific as follows:

， ；

Scattered field and the incident field of ball outside are defined as respectively: e _i , h _i with es, hs, expression formula is following (referring to Z.F.LinandS.T.Chui. " Electromagneticscatteringbyopticallyanisotropicmagneticp article.”PhysicalReviewE,vol.69,pp.056624-2-056624-24,2004)

(25)

(26)

(27)

(28)

Wherein, , for the specific inductive capacity in vacuum, for the magnetic permeability in vacuum; represent incident wave direction, polarization characteristic equivalent; In formula 27,28, subscript (3) represents that spherical vector wave functions is made up of Bessel function of the third kind; Ball interior medium and the outer medium of ball are all perfect medium, so spherome surface does not exist surface charge and surface current, so the tangential component of the Electric and magnetic fields of any point is continuous print on spherome surface, that is:

(29)

(30)

Formula 23 ~ 28 is updated to above formula 29,30 abbreviation to obtain:

Wherein, , for the radius of spheroid; , for spheric Bessel function, for first kind ball Hankel function. it is right to represent differentiate, in like manner .

Above formula is write as the form of matrix:

(31)

(32)

Solving equations 31,32 obtains:

The feature of target reflected radar scattering of wave rate is characterized by RCS (radarcrosssection.RCS), using its most basic parameter as evaluation objective Electromagnetic Scattering Characteristics, specific as follows (referring to: Z.F.LinandS.T.Chui. " Electromagneticscatteringbyopticallyanisotropicmagneticp article " PhysicalReviewE, vol.69, pp.056624-2-056624-24,2004):

(33)

(34)

(35)

Wushu 27 is updated in formula 35 and obtains:

(36)

Beneficial effect of the present invention is as follows:

The analytic solution of Bianisotropic medium ball electromagnetic scattering are proposed based on spherical vector wave functions.First time provides result of calculation.Need the determinant of the matrix equation solved containing parameter, solve parameter by determinant, then obtain the untrivialo solution of matrix equation.Owing to needing the matrix of high-order when calculating electric larger-size spheroid, and the Parameters Computer of the matrix equation of the containing parameter of high-order is not easy to calculate.So this method compares the electromagnetic scattering being applicable to solve the less Bianisotropic medium ball of electric size.

Accompanying drawing explanation

Fig. 1 is RCS and the scattering angle corresponding relation that the embodiment of the present invention 1 gives medium ball

Fig. 2 is RCS and the scattering angle corresponding relation that the embodiment of the present invention 2 gives medium ball

Fig. 3 is that the embodiment of the present invention 3 gives the RCS of medium ball and scattering angle corresponding relation Fig. 4 is RCS and the scattering angle corresponding relation that the embodiment of the present invention 4 gives medium ball

Fig. 5 is that the embodiment of the present invention 5 major parameter is the same with in example 4, research on the impact of E face RCS during change:

Fig. 6 is that the embodiment of the present invention 6 major parameter is the same with in example 4, research on the impact of H face RCS during change:

Embodiment

Below in conjunction with drawings and Examples, the invention will be further described.

A method for accurate Calculation Bianisotropic medium ball electromagnetic scattering, comprises the following steps:

Assignment is carried out to each tensor of medium eigen[value:

Calculated by fortran , and export these two matrixes to MATLAB

Determinant is calculated in MATLAB

。Draw an equation about k, solve the unknown quantity k of this equation.The solution obtained is designated as , will substitute in equation and obtain .Then pass through with calculate .

In sum, medium ball inside can both calculate, and incident field is known, by boundary condition, incident field and scattered field can be calculated the scattered field of medium ball, and then by just obtaining RCS (RCS) by scattered field

。

As shown in Figure 1, for the embodiment of the present invention 1 gives RCS and the scattering angle corresponding relation of medium ball in figure, its parameter is , , , , , the data contrasted come from document (You-LinGeng. " Scatteringofaplanewavebyananisotropicferrite-coatedcondu ctingsphere " IETMicrow.AntennasPropag, 2008,2, (2), pp.158-162.).

As shown in Figure 2, for the embodiment of the present invention 2 gives RCS and the scattering angle corresponding relation of medium ball in figure, its parameter is , , , , .

As shown in Figure 3, for the embodiment of the present invention 3 gives RCS and the scattering angle corresponding relation of medium ball in figure, its parameter is .

As shown in Figure 4, for the embodiment of the present invention 4 gives RCS and the scattering angle corresponding relation of medium ball in figure, its parameter is .

As shown in Figure 5, be that the embodiment of the present invention 5 major parameter is the same with in example 4 in figure, research on the impact of E face RCS during change:

As shown in Figure 6, be that the embodiment of the present invention 6 major parameter is the same with in example 4 in figure, research on the impact of H face RCS during change.

Claims (1)

1. a method for accurate Calculation Bianisotropic medium ball electromagnetic scattering, is characterized in that comprising the steps:

Step 1. utilizes passive maxwell equation group to derive the differential equation about magnetic induction density B with the eigen[value of two anisotropy medium;

The factor relevant with B in the differential equation is expressed with the form of spherical vector wave functions by step 2., then spherical vector wave functions M is utilized, the property of orthogonality of N draws the matrix equation of a containing parameter, the condition first utilizing matrix equation to meet untrivialo solution calculates the parameter of this matrix equation, then parameter generation is got back to the untrivialo solution obtaining matrix equation in the matrix equation of containing parameter;

Step 3. constructs a new function, uses new function V _lagain represent magnetic induction density B, and then obtain the electromagnetic field of medium ball inside, then the electromagnetic field in medium ball and the incident electromagnetic field outside ball, scattering field are updated in boundary condition, draw scattering matrix;

In described step 1, become two anisotropy medium by adding one in anisotropy medium eigen[value, the eigen[value of two anisotropy medium is specific as follows:

$\begin{array}{cc}D=\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}\&CenterDot;E+\stackrel{\&OverBar;}{\mathrm{\&xi;}}\&CenterDot;H& B=\stackrel{\&OverBar;}{\mathrm{\&mu;}}\&CenterDot;H---\end{array}\left(1\right)$

$\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}={\mathrm{\&epsiv;}}_s\left[\begin{array}{ccc}{\mathrm{\&epsiv;}}_t& -{\mathrm{i\&epsiv;}}_g& 0\\ {\mathrm{i\&epsiv;}}_g& {\mathrm{\&epsiv;}}_t& 0\\ 0& 0& 1\end{array}\right],\stackrel{\&OverBar;}{\mathrm{\&xi;}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& \mathrm{\&xi;}\end{array}\right],\stackrel{\&OverBar;}{\mathrm{\&mu;}}={\mathrm{\&mu;}}_s\left[\begin{array}{ccc}{\mathrm{\&mu;}}_t& -{\mathrm{i\&mu;}}_g& 0\\ {\mathrm{i\&mu;}}_g& {\mathrm{\&mu;}}_t& 0\\ 0& 0& 1\end{array}\right]$

Wherein, electric displacement vector D, electric field strength E, magnetic field intensity H and magnetic induction density B are all vectors, represent vector with black runic; I represents imaginary unit; ε _s, ε _t, ε _g, μ _s, μ _t, μ _gbe used to the parameter weighing medium electromagnetic property;

Passive maxwell equation group is specific as follows:

$\&dtri;\&times;E=i\mathrm{\&omega;}B---\left(2a\right)$

$\&dtri;\&times;H=-i\mathrm{\&omega;}D---\left(2b\right)$

$\&dtri;\&CenterDot;B=0---\left(2c\right)$

$\&dtri;\&CenterDot;D=0---\left(2d\right)$

Wushu 1 is updated in formula 2a, 2b, 2c, 2d, and the differential equation deriving magnetic induction density B is as follows:

$\&dtri;\&times;\&lsqb;{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}^{-1}{\mathrm{\&epsiv;}}_s\&CenterDot;(\&dtri;\&times;{\mathrm{\&mu;}}_s{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}^{-1}\&CenterDot;B)\&rsqb;+i\mathrm{\&omega;}\&dtri;\&times;\&lsqb;\stackrel{\&OverBar;}{\mathrm{\&xi;}}\&CenterDot;B\&rsqb;-k_s^{2}B=0---\left(3\right)$

Wherein, symbol × represent and curl is asked to a vector; ω is electromagnetic frequency; for inverse; $k_s^2={\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_s{\mathrm{\&mu;}}_s;$

In described step 2, by formula 3 b is write as the form of spherical vector wave functions, specific as follows:

$B=\underset{n=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{m=-n}{\overset{+n}{\&Sigma;}}{\stackrel{\&OverBar;}{E}}_{mn}\&lsqb;d_{mn}{M}_{mn}^{\left(1\right)}(k,r)+c_{mn}{N}_{mn}^{\left(1\right)}(k,r)\&rsqb;---\left(4\right)$

${\mathrm{\&epsiv;}}_s{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}^{-1}\&CenterDot;(\&dtri;\&times;{\mathrm{\&mu;}}_s{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}^{-1}\&CenterDot;B)=k\underset{n=0}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{m=-n}{\overset{+n}{\&Sigma;}}{\stackrel{\&OverBar;}{E}}_{mn}({\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}}_{mn}{M}_{mn}^{\left(1\right)}+{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}}_{mn}{N}_{mn}^{\left(1\right)}+{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{w}}}_{mn}{L}_{mn}^{\left(1\right)})---\left(5\right)$

$\stackrel{\&OverBar;}{\mathrm{\&xi;}}\&CenterDot;B=\underset{n=0}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{m=-n}{\overset{+n}{\&Sigma;}}{\stackrel{\&OverBar;}{E}}_{mn}({\hat{d}}_{mn}{M}_{mn}^{\left(1\right)}+{\hat{c}}_{mn}{N}_{mn}^{\left(1\right)}+{\hat{w}}_{mn}{L}_{mn}^{\left(1\right)})---\left(6\right)$

(4-6) expansion coefficient in formula is respectively:

$\begin{array}{cc}{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}}_{mn}=\underset{q,p}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{pq}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{p}}_{mn}^{pq}{\stackrel{\&OverBar;}{d}}_{pq}+{\tilde{p}}_{mn}^{pq}{\stackrel{\&OverBar;}{c}}_{pq}),& {\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}}_{mn}=\underset{q,p}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{pq}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{O}}_{mn}^{pq}{\stackrel{\&OverBar;}{d}}_{pq}+{\tilde{O}}_{mn}^{pq}{\stackrel{\&OverBar;}{c}}_{pq})\end{array}---\left(7\right)$

$\begin{array}{cc}{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{w}}}_{mn}=\underset{q,p}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{pq}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{q}}_{mn}^{pq}{\stackrel{\&OverBar;}{d}}_{pq}+{\tilde{q}}_{mn}^{pq}{\stackrel{\&OverBar;}{c}}_{pq}),& {\stackrel{\&OverBar;}{d}}_{mn}=\underset{v=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{u=-v}{\overset{+v}{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\tilde{g}}_{mn}^{uv}{d}_{uv}+{\stackrel{\&OverBar;}{g}}_{mn}^{uv}{c}_{uv})\end{array}---\left(8\right)$

${\stackrel{\&OverBar;}{c}}_{mn}=\underset{v=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{u=-v}{\overset{+v}{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\tilde{e}}_{mn}^{uv}{d}_{uv}+{\stackrel{\&OverBar;}{e}}_{mn}^{uv}{c}_{uv})---\left(9\right)$

${\tilde{g}}_{pq}^{mn}={\mathrm{\&sigma;}}_{nq}{\mathrm{\&sigma;}}_{mp}+\&lsqb;\frac{(n^2+n-m^2){\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}+{\mathrm{m\&mu;}}_g^{\&prime;}}{n(n+1)}\&rsqb;{\mathrm{\&sigma;}}_{nq}{\mathrm{\&sigma;}}_{mp}---\left(10\right)$

${\tilde{e}}_{pq}^{mn}=\frac{i(n+m)\&lsqb;m{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}-(n+1){\mathrm{\&mu;}}_g^{\&prime;}\&rsqb;{\mathrm{\&sigma;}}_{n-1,q}{\mathrm{\&sigma;}}_{mp}}{n(2n+1)}+\frac{i(n-m+1)\&lsqb;m{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}+{\mathrm{n\&mu;}}_g^{\&prime;}\&rsqb;{\mathrm{\&sigma;}}_{n+1,q}{\mathrm{\&sigma;}}_{mp}}{(n+1)(2n+1)}---\left(11\right)$

${\stackrel{\&OverBar;}{g}}_{pq}^{mn}=-\frac{i(n+m)(n+1)\&lsqb;m{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}+(n-1){\mathrm{\&mu;}}_g^{\&prime;}\&rsqb;{\mathrm{\&sigma;}}_{n-1,q}{\mathrm{\&sigma;}}_{mp}}{n(n-1)(2n-1)}-\frac{in(n-m+1)\&lsqb;m{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}-(n+2){\mathrm{\&mu;}}_g^{\&prime;}\&rsqb;{\mathrm{\&sigma;}}_{n+1,q}{\mathrm{\&sigma;}}_{mp}}{(n+1)(n+2)(2n+1)}---\left(12\right)$

$\begin{array}{c}{\stackrel{\&OverBar;}{e}}_{pq}^{mn}=\&lsqb;1+\frac{(4n^2+4n-3){\mathrm{m\&mu;}}_g^{\&prime;}}{n(n+1)(2n-1)(2n+3)}\&rsqb;{\mathrm{\&sigma;}}_{nq}{\mathrm{\&sigma;}}_{mp}+\frac{\&lsqb;(2n^2+2n+3)m^2+(2n^2+2n-3)n(n+1)\&rsqb;{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}}{n(n+1)(2n-1)(2n+3)}{\mathrm{\&sigma;}}_{nq}{\mathrm{\&sigma;}}_{mp}\\ -\frac{(n+1)(n+m)(n+m-1)}{(n-1)(2n-1)(2n+1)}{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}{\mathrm{\&sigma;}}_{n-2,q}{\mathrm{\&sigma;}}_{mp}-\frac{n(n-m+2)(n-m+1)}{(n+2)(2n+1)(2n+3)}{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}{\mathrm{\&sigma;}}_{n+2,q}{\mathrm{\&sigma;}}_{mp}\end{array}---\left(13\right)$

Wherein: as n=q, δ _nq=1; As n ≠ q, δ _nq=0,

${\mathrm{\&mu;}}_t^{\&prime;}=\frac{{\mathrm{\&mu;}}_t}{{\mathrm{\&mu;}}_t^2-{\mathrm{\&mu;}}_g^2},{\mathrm{\&mu;}}_g^{\&prime;}=-\frac{{\mathrm{\&mu;}}_g}{{\mathrm{\&mu;}}_t^2-{\mathrm{\&mu;}}_g^2},{\mathrm{\&epsiv;}}_t^{\&prime;}=\frac{{\mathrm{\&epsiv;}}_t}{{\mathrm{\&epsiv;}}_t^2-{\mathrm{\&epsiv;}}_g^2},{\mathrm{\&epsiv;}}_g^{\&prime;}=-\frac{{\mathrm{\&epsiv;}}_g}{{\mathrm{\&epsiv;}}_t^2-{\mathrm{\&epsiv;}}_g^2}$

$\begin{array}{cc}{\stackrel{\&OverBar;}{p}}_{pq}^{mn}={\stackrel{\&OverBar;}{e}}_{pq}^{mn}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_t^{\&prime;}={\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;},{\mathrm{\&epsiv;}}_g={\mathrm{\&mu;}}_g},& {\tilde{p}}_{pq}^{mn}={\tilde{e}}_{pq}^{mn}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_t^{\&prime;}={\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;},{\mathrm{\&epsiv;}}_g={\mathrm{\&mu;}}_g}\end{array}---\left(14\right)$

$\begin{array}{cc}{\stackrel{\&OverBar;}{O}}_{pq}^{mn}={\stackrel{\&OverBar;}{g}}_{pq}^{mn}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_t^{\&prime;}={\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;},{\mathrm{\&epsiv;}}_g={\mathrm{\&mu;}}_g},& {\tilde{O}}_{pq}^{mn}={\tilde{g}}_{pq}^{mn}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_t^{\&prime;}={\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;},{\mathrm{\&epsiv;}}_g={\mathrm{\&mu;}}_g}\end{array}---\left(15\right)$

${\hat{d}}_{mn}=\underset{v=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{u=-v}{\overset{+v}{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\tilde{r}}_{mn}^{uv}{d}_{uv}+{\stackrel{\&OverBar;}{r}}_{mn}^{uv}{c}_{uv}),{\hat{c}}_{mn}=\underset{v=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{u=-v}{\overset{+v}{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\tilde{s}}_{mn}^{uv}{d}_{uv}+{\stackrel{\&OverBar;}{s}}_{mn}^{uv}{c}_{uv})---\left(16\right)$

$\begin{array}{cc}{\tilde{r}}_{pq}^{mn}={\tilde{g}}_{pq}^{mn}\&CenterDot;\mathrm{\&xi;}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_r^{\&prime;}=-1,{\mathrm{\&mu;}}_k^{\&prime;}=0},& {\tilde{s}}_{pq}^{mn}={\tilde{e}}_{pq}^{mn}\&CenterDot;\mathrm{\&xi;}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_r^{\&prime;}=-1,{\mathrm{\&mu;}}_k^{\&prime;}=0}\end{array}---\left(17\right)$

${\stackrel{\&OverBar;}{r}}_{pq}^{mn}={\stackrel{\&OverBar;}{g}}_{pq}^{mn}\&CenterDot;\mathrm{\&xi;}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_r^{\&prime;}=-1,{\mathrm{\&mu;}}_k^{\&prime;}=0},{\stackrel{\&OverBar;}{s}}_{pq}^{mn}={\stackrel{\&OverBar;}{e}}_{pq}^{mn}\&CenterDot;\mathrm{\&xi;}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_r^{\&prime;}=-1,{\mathrm{\&mu;}}_k^{\&prime;}=0}---\left(18\right)$

Wherein, with represent the same, represent spherical vector wave functions, subscript (1) represents that vector wave function is made up of first kind spheric Bessel function, and subscript m n represents the parameter in spherical vector wave functions; (4) d in formula _mn, c _mnbeing the expansion coefficient of magnetic flux density B in Bianisotropic medium, is amount to be asked, and simultaneously k is also one and treats that quantitatively r represents a vector in spherical coordinate system; Coefficient before spherical vector wave functions is determined by the tensor in medium eigen[value, and e ₀represent the field intensity of incident electric fields;

$C_{mn}={\&lsqb;\frac{2n+1}{n(n+1)}\frac{(n-m)!}{(n+m)!}\&rsqb;}^{\frac{1}{2}}$

The character of spherical vector wave functions is utilized to obtain:

$k^2{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}}_{mn}-k_s^2{d}_{mn}+i\mathrm{\&omega;}\hat{c}=0$

$k^2{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}}_{mn}-k_s^2{c}_{mn}+i\mathrm{\&omega;}\hat{d}=0$

Be expressed as follows by the form of matrix:

$\left[\begin{array}{cc}{k}^2& 0\\ 0& k^2\end{array}\right]\left[\begin{array}{c}\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}\\ \stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}\end{array}\right]+\left[\begin{array}{cc}0& i\mathrm{\&omega;}k\\ i\mathrm{\&omega;}k& 0\end{array}\right]\left[\begin{array}{c}\hat{d}\\ \hat{c}\end{array}\right]-k_s^2\left[\begin{array}{c}d\\ c\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]---\left(20\right)$

Wherein, I is unit matrix; represent other are similar; D, c are expressed as amount d to be asked _mn, c _mnmatrix, m, n, u, v, p, q represent integer; expression formula is as follows,

$\begin{array}{cc}\left[\begin{array}{c}\hat{d}\\ \hat{c}\end{array}\right]=\left[\begin{array}{cc}\tilde{R}& \stackrel{\&OverBar;}{R}\\ \tilde{S}& \stackrel{\&OverBar;}{S}\end{array}\right]\left[\begin{array}{c}d\\ c\end{array}\right]& \left[\begin{array}{c}\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}\\ \stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}\end{array}\right]=\left[\begin{array}{cc}\tilde{p}& \stackrel{\&OverBar;}{p}\\ \widetilde{\mathrm{\&theta;}}& \stackrel{\&OverBar;}{\mathrm{\&theta;}}\end{array}\right]\left[\begin{array}{c}d\\ c\end{array}\right]\end{array}$

$\begin{array}{cc}{\tilde{R}}_{mn,uv}=\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}{\tilde{r}}_{mn}^{uv}& {\tilde{S}}_{mn,uv}=\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}{\tilde{S}}_{mn}^{uv}\end{array}$

$\begin{array}{cc}{\stackrel{\&OverBar;}{R}}_{mn,uv}=\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}{\stackrel{\&OverBar;}{r}}_{mn}^{uv}& {\stackrel{\&OverBar;}{S}}_{mn,uv}=\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}{\stackrel{\&OverBar;}{s}}_{mn}^{uv}\end{array}$

${\tilde{p}}_{mn,uv}=\underset{p,q}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{p}}_{mn}^{pq}{\tilde{g}}_{pq}^{uv}+{\tilde{p}}_{mn}^{pq}{\tilde{e}}_{pq}^{uv}),{\stackrel{\&OverBar;}{p}}_{mn,uv}=\underset{p,q}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{p}}_{mn}^{pq}{\stackrel{\&OverBar;}{g}}_{pq}^{uv}+{\tilde{p}}_{mn}^{pq}{\stackrel{\&OverBar;}{e}}_{pq}^{uv})$

${\widetilde{\mathrm{\&theta;}}}_{mn,uv}=\underset{p,q}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{o}}_{mn}^{pq}{\tilde{g}}_{pq}^{uv}+{\tilde{o}}_{mn}^{pq}{\tilde{e}}_{pq}^{uv}),{\widetilde{\mathrm{\&theta;}}}_{mn,uv}=\underset{p,q}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{o}}_{mn}^{pq}{\stackrel{\&OverBar;}{g}}_{pq}^{uv}+{\tilde{o}}_{mn}^{pq}{\stackrel{\&OverBar;}{e}}_{pq}^{uv})$

Formula 20 changes following form into:

$(\left[\begin{array}{cc}{k}^2& 0\\ 0& k^2\end{array}\right]\left[\begin{array}{cc}\tilde{p}& \stackrel{\&OverBar;}{p}\\ \widetilde{\mathrm{\&theta;}}& \stackrel{\&OverBar;}{\mathrm{\&theta;}}\end{array}\right]+\left[\begin{array}{cc}0& i\mathrm{\&omega;}k\\ i\mathrm{\&omega;}k& 0\end{array}\right]\left[\begin{array}{cc}\tilde{R}& \stackrel{\&OverBar;}{R}\\ \tilde{S}& \stackrel{\&OverBar;}{S}\end{array}\right]-k_s^{2}I)\left[\begin{array}{c}d\\ c\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]---\left(21\right)$

The implication that formula (21) is expressed is: there is such parameter k and make equation have untrivialo solution, knows that the determinant that only need make formula (21) is zero, solve parameter k by matrix knowledge, and parameter k note is separated as k _l, (l=1,2,3 ...), then use k _lsubstitution formula 21 obtains the non-vanishing solution [d of equation _{mn, l}c _{mn, l}] ^-1,

New function V is constructed in described step 3 _lspecific as follows:

$V_l=-\frac{k_l}{\mathrm{\&omega;}}\underset{n=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{m=-n}{\overset{+n}{\&Sigma;}}{\stackrel{\&OverBar;}{E}}_{mn}\&lsqb;d_{mn,l}{M}_{mn}^{\left(1\right)}(k_l,r)+c_{mn,l}{N}_{mn}^{\left(1\right)}(k_l,r)\&rsqb;$

Wherein α _lfor undetermined coefficient, determined by the boundary condition of medium ball surface;

Order wherein a _lfor representing V _lweight; The corresponding magnetic field obtaining ball inside electric field the incident electric fields E of ball outside _i, magnetic field H _i, scattering electric field E _s, magnetic field H _sbe updated to as in downstream condition:

[E _I+E _s]×e _r＝E _l×e _r[H _I+H _s]×e _r＝H _l×e _r

Wherein e _rfor the direction vector of Electromagnetic Wave Propagation;

Abbreviation draws scattering matrix (21) formula after arranging, and so just calculates RCS.